AP Precalculus: Continuity – Graphs & Formulas

1. Formal Definition of Continuity

\( f(x) \) is continuous at \( x = a \) if:
- \( \lim_{x \to a} f(x) \) exists
- \( f(a) \) exists
- \( \lim_{x \to a} f(x) = f(a) \)
If any one is false, function may be discontinuous at \( x = a \)

2. Types of Discontinuity

Removable: "hole" in the graph, limit exists but \( f(a) \neq \lim_{x \to a} f(x) \)
Jump: Left and right limits exist but are different
Infinite: Graph approaches \( \pm\infty \) (asymptote); limit does not exist

3. One-Sided Continuity

Right-continuous at \( a \): \( \lim_{x \to a^+} f(x) = f(a) \)
Left-continuous at \( a \): \( \lim_{x \to a^-} f(x) = f(a) \)

4. Continuity on Intervals

Function is continuous on interval \( [a, b] \) if continuous at every point \( x \) in \( (a, b) \)
At endpoints of closed interval:
- Left-continuous at \( a \), right-continuous at \( b \)

5. Analyzing Points of Discontinuity

Examine graph or formula: test three continuity conditions at suspect points
For rational \( f(x) \): check denominator zeros, factor/removable points
For piecewise: check at boundaries